# The length of excitable knots

## TL;DR

This study uses the FitzHugh-Nagumo model to simulate long-term dynamics of knotted vortex solutions in excitable media, revealing preserved topology, minimal lengths, and boundary effects, offering a new approach to knot analysis.

## Contribution

It introduces a novel field theory approach to knot study through numerical simulations of vortex dynamics in excitable media, focusing on topology preservation and minimal length characterization.

## Key findings

- Knot topology is preserved during evolution in the FitzHugh-Nagumo model.
- Each knot has a well-defined minimal length comparable to ideal knot ropelength.
- Boundary effects play a significant role in stabilizing knot length.

## Abstract

The FitzHugh-Nagumo equation provides a simple mathematical model of cardiac tissue as an excitable medium hosting spiral wave vortices. Here we present extensive numerical simulations studying long-term dynamics of knotted vortex string solutions for all torus knots up to crossing number 11. We demonstrate that FitzHugh-Nagumo evolution preserves the knot topology for all the examples presented, thereby providing a novel field theory approach to the study of knots. Furthermore, the evolution yields a well-defined minimal length for each knot that is comparable to the ropelength of ideal knots. We highlight the role of the medium boundary in stabilizing the length of the knot and discuss the implications beyond torus knots. By applying Moffatt's test we are able to show that there is not a unique attractor within a given knot topology.